Is there the same amount of fractions as whole numbers? How can this be proved? Well, before delving into the arguments for the proof of this assertion, let's see some peculiarities of the fractions.
1. There is always a fraction between any two given fraction. This can be shown as follows: suppose we are given the two fractions a/b and c/d , then the "average" of the two fractions is in the middle between both. It can be found by the simple formula (ad + bc) /(2 bd).
For example, the fraction between 1/4 and 1/3 is (1x3 + 1x4)/(2x3x4) = 7/24. This last fraction should be exactly in the middle of 1/3 and 1/4. Fraction position are difficult to imagine in the real numbers line. For this reason, let us convert each fraction to its decimal equivalent like this: 1/3 = 0.333333..., and, 1/4 = 0.250000. But 7/24 = 0.291666... See that this number is exactly between the two previous fractions, because 0.291666... -0.250000 = 0.333333... - 0.291666...
The lesson learned so far is that between any two fractions we can indefinitely keep finding fractions and more fractions without any limitation; and this is true no matter how near the fractions are to each other. Note that the natural number series doesn't has this property: given any two natural numbers no always is there another integer between them, much less exactly in the middle of both.
Since fractions can be uniquely converted to decimals we can state the same property for decimals as for fractions: There is always a decimal between any two given decimals.
2. The fractions lack the property of one being the successor of another. Contrary to the fractions, for each natural number the next whole number one is always one unit ahead; this is a "property" we can call "of successorship": to the natural number n follows exactly one integer, an that integer is exactly n + 1. But we cannot say that the "next" fraction that follows the fraction a/b is a/b + 1; in fact by what we just seen we can say that between a/b and a/b + 1 there is an infinity of fractions. Analogously as we stated for the fractions we can say for the decimals the following: No decimal is the successor not the predecessor of any other decimal.
In view of the above arguments, we can be tempted to proclaim that there are many more fractions than natural numbers, since by the absence of the property of "successorship" of the fractions it is impossible of counting them. But the genius of George Cantor comes to rescue us from this logical swamp: we are reasoning "linearly" as in the number line, so why not reason "diagonally"?
Let's see a peculiar arrangement of all the possible positive fractions. See that starting from the top row, we are arranging all the fractions with denominator 1, then all the fractions with denominator 2, etc. In this schema, no positive fraction is left outside. Now, starting with the 1/1 fraction, if we move in the direction of the arrows, we can numerate all fractions with no fraction left out.
To enumerate the fraction, we can say that 1/1 is the "first" fraction, 2/1 is the "second" fraction, 1/2 is the "third" fraction, etc.
With this zigzag arrangement of the fractions we can see that they can be enumerated using the natural numbers; therefore, the fractions are infinitely countable.
This is a beautiful "paradox" not easy to digest. It is commonly known as Torricelli's trumpet in honor to his discoverer, but mostly known as Gabriel's horn.
Evangelista Torricelli (1608-1647) was a student of Galileo. Torricelli studied the properties of the curve given by the reciprocal function y = 1/f( x ) for the domain of the real numbers x from 1 to infinity. We all are familiar with this curve from the simple exercises of elementary algebra. This is a simple and elegant curve that continually asymptotically approaches the X-axis without ever touching or crossing it.
What is very interesting about this curve is that if the curve is rotated around the X-axis, we obtain the surface called The Torricelli's trumpet.
The modern approach toward the "Torricelli's trumpet" is to visualize this object as a solid that can be subdivided into as many circular slabs as we desire. Then compute and add the volume and surface area of each slab.
Using integration —that is, the successive addition of the infinitesimal surfaces and volumes— the volume value approaches the number 
, while the surface area grows without control to infinity. But ( = 3.1416 ...) is a finite number, while the infinity is not a number at all, therefore, how can they coexist as properties of the same "object" or entity? This is the "paradox": how can an "object" can have infinite bounding surface while at the same time be finite in its "interior" volume?
Since the Torricelli's trumpet extends indefinitely horizontally, when we rotate it downwards the trumpet becomes an infinite vase extending downward without bound making of it the infinite vase we began talking about.
Torricelli's figure is paradoxical because it has the attributes of having finite volume and infinite surface at the same time. Our intuition expects that infinite surfaces are needed to enclose infinite objects only, while at the same time our intuition dictates us that finite volumes could bounded by finite surfaces only.
However, a volume of just units can be easily contained by many types of shapes. For example, a cylinder of height and radius 1 unit contains a volume of exactly units as the Torricelli's horn does.
The formula for the volume of a cylinder is:
V = × r2 × h
where r is the radius of the cylinder and h is its height. By this formula, the volume enclosed by a "unit" cylinder is just units, as in the Torricelli's horn.
In the other hand, the surface area needed to enclose and hold this volume is not infinite. Let us take the lateral curved surface and forget for now the top and bottom faces. The area of this round face is just
S = 2××r×h
so the area is 2 × .
Note that if this cylinder is stretched down following the inverse function as Torricelli did, we will obtain the same "horn" as Torricelli obtained: the volume will be preserved but the rounded lateral are will become infinite.
Go to More myths in Part 1 of this article.
See also in this Website the article: Where does infinity begin?
See also in this Website the article: Three unexpected behaviors of the infinite
